A335358 a(n) is the X-coordinate of the n-th point of the Koch curve; sequence A335359 gives Y-coordinates.

0, 1, 1, 2, 3, 3, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 8, 9, 9, 8, 7, 7, 6, 7, 7, 8, 9, 9, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 26, 27, 27, 26, 25, 25, 24

Offset

0,4

Comments

Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:

Y

/

/

0 ---- X

We can build the Koch curve as follows:

- start from the origin,

- for k = 0, 1, ...:

- compute t as the number of 1's minus the number of 2's

in the base-4 representation of n (t = A065359(n)),

- move one step forward with an angle of t * 60 degrees

with respect to the X-axis.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Artem Litvinov, A study of the Koch polyline, 2021 [in Russian].

Wikipedia, Koch snowflake

Index entries for sequences related to coordinates of 2D curves

Formula

From Andrey Zabolotskiy, Nov 12 2021: (Start)

a(2*n) = a(n) + 2*y(n),

y(2*n) = a(n) - y(n),

where y(k) = A335359(k). See Litvinov, 2021. (End)

Example

The Koch curve starts (on a hexagonal lattice) as follows:
   .   .   .   .   .   .   +   .   .   .   .   .   .
                          /8\
     .   .   .   .   +---+   +---+   .   .   .   .
                     6\  7   9  /10
   .   .   .   +   .   +   .   +   .   +   .   .   .
              /2\     /5        \     / \
     .   +---+   +---+   .   .   +---+   +---+   .
         0   1   3   4           12  13  15  16
Hence, a(4) = a(5) = a(7) = a(8) = 3.

Prog

(PARI) { hex = [1, I, I-1, -1, -I, 1-I]; z=0; for (n=0, 72, print1 (real(z)", "); q=digits(n, 4); d=sum(k=1, #q, if (q[k]==1, +1, q[k]==2, -1, 0)); z+=hex[1+d%#hex]) }

Crossrefs

Cf. A065359, A335359.

Sequence in context: A025076 A110006 A289831 * A296611 A308661 A286104

Adjacent sequences: A335355 A335356 A335357 * A335359 A335360 A335361

Keyword

nonn,look,base

Author

Rémy Sigrist, Jun 03 2020

Status

approved